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Mirrors > Home > MPE Home > Th. List > Mathboxes > imaiun1 | Structured version Visualization version GIF version |
Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.) |
Ref | Expression |
---|---|
imaiun1 | ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 3377 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) | |
2 | vex 3354 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | elima3 5613 | . . . . 5 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
4 | 3 | rexbii 3189 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
5 | eliun 4659 | . . . . . . 7 ⊢ (〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵) | |
6 | 5 | anbi2i 609 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵)) |
7 | r19.42v 3240 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵)) | |
8 | 6, 7 | bitr4i 267 | . . . . 5 ⊢ ((𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
9 | 8 | exbii 1924 | . . . 4 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐵)) |
10 | 1, 4, 9 | 3bitr4ri 293 | . . 3 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶)) |
11 | 2 | elima3 5613 | . . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) |
12 | eliun 4659 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶)) | |
13 | 10, 11, 12 | 3bitr4i 292 | . 2 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶)) |
14 | 13 | eqriv 2768 | 1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 = wceq 1631 ∃wex 1852 ∈ wcel 2145 ∃wrex 3062 〈cop 4323 ∪ ciun 4655 “ cima 5253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-iun 4657 df-br 4788 df-opab 4848 df-xp 5256 df-cnv 5258 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 |
This theorem is referenced by: trclimalb2 38544 |
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